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Author(s)
Corrective Voltage Control Scheme Considering Demand
Response and Stochastic Wind Power
Rabiee, Abbas; Soroudi, Alireza; Mohammadi-Ivatloo,
Behnam; Parniani, Mostafa
Publication
date
2014-11
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information
IEEE Transactions on Power Systems, 29 (6): 2965-2973
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Institute of Electrical and Electronics Engineers
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http://hdl.handle.net/10197/6115
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1
Corrective Voltage Control Scheme Considering
Demand Response and Stochastic Wind Power
Abbas Rabiee, Alireza Soroudi, Member, IEEE, Behnam Mohammadi-ivatloo, Member, IEEE, and Mostafa Parniani,
Senior Member, IEEE
Abstract—This paper proposes a new approach for corrective
voltage control (CVC) of power systems in presence of uncertain
wind power generation and demand values. The CVC framework
deals with the condition that a power system encounters voltage
instability as a result of severe contingencies. The uncertainty
of wind power generation and demand values is handled using
scenario-based modeling approach. One of the features of the
proposed methodology is to consider participation of demand-side
resources as an effective control facility that reduces control costs.
Active and reactive re-dispatch of generating units and involuntary
load curtailment are employed along with the voluntary demandside participation (demand response) as control facilities in the
proposed CVC approach. The CVC is formulated as a multiobjective optimization problem. The objectives are ensuring a
desired loading margin while minimizing the corresponding control costs. This problem is solved using ǫ-constraint method, and
fuzzy satisfying approach is employed to select the best solution
from the Pareto optimal set. The proposed control framework
is implemented on the IEEE 118-Bus system to demonstrate its
applicability and effectiveness.
Keywords—Demand response (DR), loading margin (LM),
scenario-based approach, voltage security, wind power generation,
voltage control.
N OMENCLATURE
A. Sets:
N BCV C
N GCV C
NG
N Gb
NS
NB
NL
Set of buses selected for the CVC program.
Set of generating units that participate in the
CVC program.
Set of generating units.
Set of generating units located at bus b.
Set of scenarios.
Set of system buses.
Set of transmission lines.
Magnitude/angle of bj th element of admittance matrix at the post-contingency
state.
(P/Q)G
Max/minimum active/reactive power of
i,max / min
generator i.
G,up/down
∆(P/Q)i,max
Maximum active power inc/decrement for
generator i.
DR/ILC
∆(P/Q)b,max
Maximum active/reactive power decrement in DR/ILC program at bus b.
Sℓmax
Maximum transfer capacity of line ℓ.
Qw
Max/minimum reactive power output of
b,max / min
wind turbine at bus b
max / min
Vb
Max/minimum voltage at bus b.
vm
Mean wind speed in m/s.
πs
Probability of scenario s.
P/Q,up/down
µi
Price offered by generator i to
inc/decrease its active/reactive power
schedule.
P/Q,DR/ILC
Price offered by demand b to decrease
µb
its active/reactive power schedule in the
context of DR/ILC program.
wpb,s
Percent of wind turbine capacity produced at bus b and scenario s.
KG,i
Rate of change in active power generation
of unit i.
KD,b
Rate of load change at bus b.
w
Pb,r
Rated active power of wind turbine connected to bus b.
(P/Q)G,sch
Scheduled active/reactive power of geni
erator i.
v
Wind speed in m/s.
Ybj /φbj
D. Variables:
B. Indices:
i
s
b
ℓ
C. Parameters:
(P/Q)D
b,s
λdes
G,up/down
Index
Index
Index
Index
for
for
for
for
generation units.
scenarios.
system buses.
transmission lines.
Active/reactive power consumption of
load connected to bus b at scenario s.
Desired loading margin.
A. Rabiee is with the Department of Electrical Engineering, Faculty of Engineering,
University of Zanjan, Zanjan, Iran, (e-mail: rabiee@znu.ac.ir).
A. Soroudi (e-mail: alireza.soroudi@gmail.com).
B. Mohammadi-ivatloo is with the Faculty of Electrical and Computer Engineering,
University of Tabriz, Tabriz, Iran (e-mail: mohammadi@ieee.org).
M. Parniani is with the Center of Excellence in Power System Control and Management
(CEPSCM), Department of Electrical Engineering, Sharif University of Technology (SUT),
Tehran, Iran (e-mail: parniani@sharif.edu).
∆Pi
w
Pb,s
/Qw
b,s
PiG
D
P̂b,s
/Q̂D
b,s
G
P̂i,s
/Q̂G
i,s
DR/ILC
∆(P/Q)b,s
Sℓ,s (.)
λs
G,up/down
∆Qi,s
QG
i,s
Active power inc/decrement of generator i.
Active/reactive wind power production injected to bus b in scenario s.
Active power production of generator i.
Active/reactive power consumption of load
connected to bus b in scenario s at loadability limit point.
Active/reactive power production of generator i in scenario s at loadability limit.
Active/reactive power decrement in
DR/ILC program at bus b in scenario s.
Apparent power of line ℓ in scenario s.
Loading margin in scenario s.
Reactive power inc/decrement of generator
i in scenario s.
Reactive power production of generator i
in scenario s.
2
Vb,s /θb,s
V̂b,s /θ̂b,s
Voltage magnitude/angle of bus b in scenario s.
Voltage magnitude/angle of bus b in scenario s at loadability limit point.
I. I NTRODUCTION
N recent years voltage instability has received wide attention
among power system utilities, due to the several reported
incidents caused by this phenomenon [1], [2]. The growth of
electrical energy demand, economic and environmental concerns in expanding generation and transmission capacities,
and market pressure to reduce operating costs have forced
power systems to operate ever closer to their voltage stability
limits. Under such circumstances, there is possibility of voltage
instability occurrence, and therefore, it has to be considered as
an integral part of power system operation and planning studies.
Also, the recent trends toward smart grids and increasing share
of renewable energy resources in many power systems, have
intensified the needs for powerful approaches for power system
security enhancement [3], [4].
In order to restore voltage stability of power system, one had
to curtail some of the system loads in case of heavy system
loadings or occurrence of critical contingencies [5]. Nevertheless, forced load curtailment is undesirable for customers and
the system operator should pay high penalties denoted as value
of lost load (VLL). Demand Response (DR) program can be
a good alternative for involuntary load curtailment (ILC), by
curtailing the customer loads with their permissions. DR is
defined as changes of customer loads from nominal value in
response to price changes, incentive payments of operator or
reliability problems [6], [7]. Beside the financial benefits of DR
for customers (bill savings and incentive payments) and other
market participants (lowering market clearing price and capacity
requirement), DR program can be utilized for enhancing power
system reliability and stability. The impact of DR program
on power system reliability is investigated in [8]. Application
of DR in enhancing frequency stability of power system is
studied in [9] and [10]. Using DR programs for improvement
of small-signal and transient stability of power systems with
high wind power penetration are proposed in [11] and [12],
respectively. An event-driven emergency DR scheme to enhance
power system security has been proposed in [13]. With the
increasing growth of wind power penetration in power systems,
the effect of wind power in reactive power control is studied
in [14], [15]. A new index titled reactive power loadability (Qloadability) is proposed in [14] for finding the optimal location
of reactive power compensation devices in distribution systems
considering different wind power penetration levels. The effect
of emergency demand response program (EDRP) and time of
use program (TOU) programs on operating cost of a wind
integrated distribution network is studied in [15]. An overview
of the classic and advanced voltage control schemes along with
voltage control practices around the world are provided in [16].
I
Corrective voltage control (CVC) is initiated in conditions
that the system encounters voltage instability as a result of
severe contingencies [17]. In this situation, CVC actions bring
the post-contingency operating point to an equilibrium point
with a sufficient loading margin (LM), immediately after occurrence of a contingency. LM is defined as the amount of
load increase not arousing voltage instability or violation of
operational constraints. An operating point is secure if its LM
is more than a desired positive value. Otherwise, the system is
insecure. Moreover, if the LM is negative (i.e the load demand
is greater than the power generation), the system is unstable
[18].
Satisfying power systems voltage security in the framework
of preventive/corrective control is not a new problem and has
been investigated in the literature [19], [18]. For instance, [19],
after bringing an unstable operating point to a stable region
in the context of CVC, first tries to satisfy a desired voltage
stability margin, and then brings the operating point to a secure
region where the operational constraints are satisfied. But, the
proposed formulation in this paper satisfies both the voltage
stability margin and the operational constraints in one step and
in one optimization problem, resulting in better solutions. In
[18], in spite of thoroughness of the technical work, the problem
of controlling voltage security is not presented in a hierarchical
framework to recognize what types of control measures are to be
taken in different threatening conditions. Also, its formulation
does not exploit load shedding as a fast and effective tool for
providing voltage security.
This paper presents CVC as an optimization problem, considering the complete nonlinear model of the system; and
hence, eliminates the above-mentioned problems with [19]. In
contrast to [18], it provides further insight toward controlling
voltage security in power systems, both technically and economically. It also uses probabilistic load shedding in the form
of DR for CVC, which has not been addressed before in the
literature [20]. In addition, noting the increasing penetration
of wind power generation in nowadays power systems, scenario based approach [21] is adopted for appropriate modeling
of intermittent wind power generation in the proposed CVC
model. The proposed CVC approach is formulated as a multiobjective optimization problem. The objectives are maximizing
LM and minimization of its corresponding CVC cost. This
multi-objective problem is solved using ǫ-constraint method and
the Pareto optimal set is obtained. Then, by employing fuzzy
satisfying approach, the best solution is selected from this set.
Given the above context, the contributions of this paper are:
1) To assess the effect of the intermittent wind power generation on CVC using a scenario based approach
2) To consider the total nonlinear model of the power system
in optimized CVC
3) To model technical and economical aspects simultaneously
by proposing a multi-objective optimization framework
4) To model the DR programs and customer choices in the
required load shedding for CVC
5) To utilize ǫ-constraint method and fuzzy satisfying approach for solving and selecting the best compromising
solution of the multi-objective optimization problem.
The rest of this paper is set out as follows. Section II
describes CVC procedure. Section III presents the utilized
uncertainty modeling approach, the problem formulation and
its solution methodology. Simulation results are presented in
Section IV. Finally, the findings of this work are summarized
in Section V.
II.
C ORRECTIVE VOLTAGE C ONTROL (CVC)
Based on the experience of Western Electricity Coordinating
Council (WECC) [22], for secure operation of a power system
from voltage security point of view, it is suggested to preserve
specified LMs for both the base case and post-contingency
conditions. If the system is unstable as a result of a severe
contingency, fast control actions should be taken in order to
3
prevent voltage instability and provide voltage security. This
kind of control is referred as corrective voltage control or
emergency voltage control [23]. In this regard, the mission
of CVC is to improve the LM from a negative value to a
desired post-contingency value (λdes ), using fast remedial
actions. These remedial controls are active power generation
re-dispatch of fast-response generating units, reactive power
generation re-dispatch of all dynamic VAR sources including
synchronous generators and condensers, FACTS controllers’
settings, switching of fast switchable capacitor banks/reactors,
and load curtailment. To explain the CVC with more details,
consider Fig. 1, which depicts PV curves of an arbitrary load
bus for three states as follows:
Pre-contingency (curve (1))
Post-contingency - before applying CVC (curve (2))
Post-contingency - after applying CVC (curve (3))
The pre-contingency operating point A (with the demand
of PL0 ) is located on curve (1). After occurrence of a severe
contingency, the PV curve changes to curve (2). Thus, the
LM becomes negative and the post-contingency equilibrium
vanishes. Implementing the CVC will change the PV curve
from curve (2) to curve (3); and hence the new operating point
B is achieved, which is a stable and secure post-contingency
equilibrium point. The loading parameter, λdes , shown in Fig.
1, indicates the desired LM which should be ensured by
implementing the CVC.
V
(1)
(3)
A
(2)
B
B′
λdes
P
PL0
Fig. 1.
Evolution of the operating points during the CVC step
III. P ROBLEM FORMULATION
A. Uncertainty modeling of wind power and electric demand
The variation of wind power generation is an uncertain parameter which can be modeled probabilistically using historical
data records of wind speed [21], [24]. In this paper, variation of
wind speed, v, is modeled using Rayleigh probability density
function (PDF) [25].
v2
v
)
exp[−(
)]
(1)
c2
2c2
The generated power of a wind turbine in terms of wind speed
is approximated as follows [24]:

c
c
if v ≤ vin
or v ≥ vout

0
c
v−vin
w
c
w
(2)
Pb (v) = vc −vc Pb,r if vin ≤ v ≤ vrated
in

 rated
w
Pb,r
else
P DF (v) = (
c
c
c
are the cut-in, rated and cut-off
where vin
, vrated
, and vout
w
speed of wind turbine, respectively. Pb,r
denotes rated power
of the wind turbine installed at bus b. More accurate relations
could also be used instead of the linear P − V relation for the
c
interval vin
≤ v ≤ vrated [26]. Using the technique described
in [24], [27], the PDF of wind speed is divided into several
intervals, and the probability of falling into each interval is
calculated. Each interval is given a mean value which is further
used. Demand values are modeled using a normal distribution
function with a known mean and variance. It is assumed that
the load and wind power generation scenarios are independent
so the scenarios are combined to construct the whole set of
scenarios as follows [28].
πs = πw × πl
(3)
where πw and πl are the probabilities of the w-th wind and the
l-th load scenarios, respectively. The total number of scenarios,
i.e., N S, will be ln × wn , where wn , ln are the number of wind
and load states.
B. Formulation of the proposed CVC
The goal of the system operator is to optimize the expected
values of two objective functions, namely, minimizing the
cost of corrective voltage control and maximizing the loading
margin of each scenario, while satisfying network’s equality
and inequality operational constraints. The equality constraints
include AC power flow equations, and the inequality constraints
consist of the limits of system variables (e.g. voltages, active
and reactive powers, etc.). As noted in [18], the constraints
should be considered for both the post-contingency operating
point (i.e. point B in Fig. 1) and its corresponding loadability
limit point (i.e. point B ′ in Fig. 1) to ensure the relation between
the operating point and the critical point. Also, determination of
control measures considering merely the operational constraints
at the critical point may cause voltage violation in operating
points with lower load levels [20]. Thus, the problem is inherently a multi-objective optimization problem. The vector of
objective functions is described as follows.
min f¯(u¯s , x¯s , y¯s , s)
(4)
¯
f (u¯s , x¯s , y¯s , s) = [f1 (u¯s , x¯s , y¯s , s), −f2 (u¯s , x¯s , y¯s , s)]
where,
X P,up/down
G,up/down
(5)
∆Pi
µi
f1 (u¯s , x¯s , y¯s , s) =
i∈N G
CV C
 P
Q,up/down
G,up/down

∆Qi,s
 P
i∈N G µi
X
P,DR/ILC
DR/ILC
+
πs
+ i∈N BCV C µb
∆Pb,s

P

DR/ILC
Q,DR/ILC
s∈N S
∆Qb,s
+ i∈N BCV C µb
X
f2 (u¯s , x¯s , y¯s , s) =
πs λ s





(6)
s∈N S
The negative sign before f2 in right hand side of (4),
maximizes f2 . Equation (5) is the expected cost of CVC. The
first line in (5) represents the cost of active power re-dispatch
of fast-response units. The second line is the expected cost
of generating units’ reactive power re-dispatch. Also, the third
and the forth lines include the expected costs of active and
reactive load curtailment performed by DR program, and the
cost of ILC, respectively. The expected value of LM is given
by (6). Besides, u¯s , x¯s , y¯s are the vectors of control, state and
dependent variables at the post-contingency operating point in
scenario s, respectively. Detailed description of these variables
4
will be given later in this paper. The objective function in (4) is
subject to the following constraints: For ∀b ∈ N B, ∀s ∈ N S:
!
N
Gb
X
w
D
DR
ILC
PiG + Pb,s
=
(7)
− Pb,s
− ∆Pb,s
− ∆Pb,s
i=1
NB
X
Vb,s
Vj,s Ybj cos(θb,s − θj,s − φbj )
j=1
N
Gb
X
QG
i,s
i=1
NB
X
Vb,s
!
D
DR
ILC
=
+ Qw
b,s − Qb,s − ∆Qb,s − ∆Qb,s
(8)
Vj,s Ybj sin(θb,s − θj,s − φbj )
j=1
∀i ∈ N GCV C :
PiG = PiG,sch + ∆PiG,up − ∆PiG,down
0≤
∆PiG,up ≤
∆PiG,down
G,up
∆Pi,max
G,down
≤ ∆Pi,max
(9)
(10)
0≤
∀i ∈ N G; ∀s ∈ N S :
(11)
G,sch
+ ∆QG,up
− ∆QG,down
QG
i,s = Qi
i,s
i,s
(12)
G
G
Pi,min
≤ PiG ≤ Pi,max
G
G
QG
i,min ≤ Qi,s ≤ Qi,max
(13)
(14)
∀i ∈ N G; ∀s ∈ N S, ∀b ∈ N B, ∀ℓ ∈ N L :
0 ≤ ∆QG,up
≤ ∆QG,up
i,s
i,max
(15)
0 ≤ ∆QG,down
≤ ∆QG,down
i,s
i,max
DR,max
DR
0 ≤ ∆Pb,s
≤ ∆Pb,s
DR,max
0 ≤ ∆QDR
b,s ≤ ∆Qb,s
ILC,max
ILC
0 ≤ ∆Pb,s
≤ ∆Pb,s
ILC,max
0 ≤ ∆QILC
b,s ≤ ∆Qb,s
Vbmin ≤ Vb,s ≤ Vbmax
|Sℓ,s (V, θ, s)| ≤ Sℓmax
(16)
∀b ∈ N B; ∀s ∈ N S :
!
N
Gb
X
G
w
D
P̂i,s
+ Pb,s
− P̂b,s
=
(17)
(18)
(19)
(20)
(21)
(22)
(23)
i=1
V̂b,s
NB
X
V̂j,s Ybj cos(θ̂b,s − θ̂j,s − φbj )
j=1
N
Gb
X
i=1
Q̂G
i,s
!
D
+ Qw
b,s − Q̂b,s =
(24)
NB
X
Finally ∀b ∈ N BG , ∀s ∈ N S, ∀i ∈ N G:
G
G
G
Pi,min
≤ P̂i,s
≤ Pi,max
(30)
G
G
QG
i,min ≤ Q̂i,s ≤ Qi,max
Vbmin ≤ V̂b,s ≤ Vbmax
(31)
(32)
V̂b,s = Vb,s = Vb
λs ≥ λdes > 0
(33)
(34)
Constraints (7)-(22) correspond to the post-contingency secure operating point (point B in Fig. 1), whereas (23)-(32)
correspond to the post-contingency loadability limit point (point
B́ in Fig. 1). It is worth mentioning that (33) guarantees
feasibility of the post-contingency operating point (point B in
Fig. 1) and a trajectory from point B leading to point B́ in effect
of load increment [29]. Also, (34) ensures the desired LM (i.e.
λdes ) for all scenarios. The desired LM is set by the network
operator. The sets of control, state and dependent variables are
described as follows.


Vb
b ∈ NG
 λs

s ∈ NS


 ∆PiG,up/down

i
∈
N
G
CV C

 (35)
ūs =  P w , Qw

b
∈
N
B
;
s
∈
N
S
w
 b,sDRb,s DR

 ∆Pb,s , ∆Qb,s
b ∈ N BCV C ; s ∈ N S 
ILC
∆Pb,s
, ∆QILC
b ∈ N BCV C ; s ∈ N S
b,s
x̄s = Vb,s , V̂b,s , θb,s , θ̂b,s b ∈ N B, s ∈ N S
(36)
G,up/down
i ∈ N G, s ∈ N S
, Q̂G
i,s
ȳs = ∆Qi,s
(37)
Sℓ,s
ℓ ∈ N L, s ∈ N S
The proposed model utilizes a two-stage stochastic modeling
technique. In this approach, the decision variables are divided
into two different categories, namely, here and now & wait
and see. The values of wait and see variables differ from one
scenario to another, while the values of here and now variables
are the same for all scenarios. This means that the here and now
decisions are made prior to realization of uncertain parameters
and the wait and see variables are calculated to be applied
posterior to the realization of uncertain parameters (i.e. after
realization of the corresponding scenario). The type of the
decision variables are identified in the following:
Here and now decision variables (DHN ):
o
n
G,up/down
(38)
DHN ∈ ∆Pi
, Vb
Wait and see decision variables (DW S ):


w
w

 Pb,s , Qb,s , λs
DR
∆Pb,s
, ∆QDR
DW S ∈
b,s


ILC
∆Pb,s
, ∆QILC
b,s
(39)
V̂j,s Ybj sin(θ̂b,s − θ̂j,s − φbj ) C. Solution Procedure
Various methods are available to solve multi-objective optimization problems such as weighted sum approach, ǫ-constraint
method, evolutionary algorithms, etc [30]. In this paper, the
G
(25)
P̂i,s
= min PiG,max , (1 + KG,i λs ) PiG
proposed multi-objective model of the CVC is solved using
ǫ-constraint method, which is an efficient technique to solve
D
D
DR
ILC
(26)
P̂b,s = (1 + KD,b λs ) Pb,s − ∆Pb,s − ∆Pb,s
problems with non-convex Pareto front. This method generates
D
DR
ILC
single objective subproblems, by transforming all but one objec(27)
Q̂D
=
(1
+
K
λ
)
Q
−
∆Q
−
∆Q
D,b
s
b,s
b,s
b,s
b,s
w
w
tive into constraints. The upper bounds of these constraints are
0 ≤ Pb,s ≤ wpb,s ∗ Pb,r
(28)
given by the epsilon-vector and by varying it, the Pareto front
w
w
w
(29)
Qb,min ≤ Qb,s ≤ Qb,max
can be obtained. The concept of Pareto optimality is explained
V̂b,s
j=1
5
TABLE I.
T HE WIND - LOAD SCENARIOS AND THEIR PROBABILITES
s1
s2
s3
s4
s5
s6
s7
s8
s9
load(%)
98
100
102
98
100
102
98
100
102
wind(%)
100
100
100
50
50
50
0
0
0
πs
0.015
0.070
0.015
0.120
0.560
0.120
0.015
0.070
0.015
700
600
500
400
300
200
100
0
Generator Bus "umber
Fig. 2.
Initial dispatch of active power generations (MW)
0.35
0.3
Loading margin
IV. S IMULATION RESULTS
The proposed algorithm is implemented in General Algebraic
Modeling System (GAMS) [32] environment and solved by
SNOPT solver [33]. This section presents the study results
conducted on the IEEE 118-bus test system. The data of this
system is given in [34]. In order to determine the LM, the loads
are increased evenly with constant power factor characteristic.
Active powers of the generators, not hitting their upper limits
in the base-case, are also increased evenly. The costs of up and
down re-dispatching active and reactive powers of generating
units are assumed to be 125%, 25%, 12.5%, 2.5% of the base
case locational marginal price (LMP) of the buses connected
to generating units, respectively. The cost of ILC at each bus is
considered to be 100 times of LMP of that bus. The costs paid
to participants of DR programs to deploy their load reduction
in a given bus are also assumed to be 10 times of the LMP of
that bus. The desired post-contingency LM is considered to be
10%.
This study investigates a double-contingency case, i.e.
simultaneous outages of the line L1−3 (between the buses
B1 and B3 ) and the generator G5 (located at bus B10 ). This
event leads to voltage collapse in the system. Hence, the CVC
is taken to restore voltage stability, and to provide voltage
security in the post-contingency condition. The CVC improves
the LM from −8.1% to the desired post-contingency value of
10%. It is assumed that utmost 5% of the demand at buses
B44 , B45 , B47 , B48 , B50 − B53 , B57 , B58 , B82 − B84 , B86
and B88 are selected for DR program. Also, it is assumed that
the fast-response generation units are those located at buses
B24 , B25 , B46 , B49 , B54 − B56 , B85 , B87 , B89 and B90 .
It is also assumed that five wind farms exist in this system,
which are located at buses B14 , B51 , B57 , B102 and B115 . The
total wind generation capacity of each wind farm is assumed
to be 200M W . The wind and load scenarios are combined and
the overall wind-load scenarios along with the corresponding
wind/load percentages and probabilities are given in Table I.
Initial schedule of active power generations are given in Fig.
2.
In order to solve the multi-objective CVC problem by
ǫ-constraint method, maximum and minimum values of the
expected LM (i.e. f2 ) are calculated, which are equal to 0.3267
and 0.1000, respectively. These border values are achieved by
maximizing and minimizing f2 individually as the objective
function of CVC. Then, by assuming f2 as a constraint of
the CVC (in the form of f2 ≥ ǫ), lower bound of f2 (i.e.
ǫ) varies from 0.1000 to 0.3267 and f1 is minimized as the
sole objective function of CVC. Correspondingly, the Pareto
optimal front of the two objective functions is derived, which
is depicted in Fig 3. This Pareto front consists of 40 Pareto
optimal solutions.
Table II shows the values of both objective functions for all
40 Pareto optimal solutions. Among these optimal solutions,
Solution#1 is the minimum cost case, with the cost equals
to $10431.511 and the LM of 10%. Also, Solution#40 is the
maximum LM case, where the LM is 0.3267 and the CVC cost
is $841307.979. Active power redispatch of the fast-response
generating units are give in Fig. 4. For this solution, the wind
power scenario-based active and reactive power dispatchs are
given in Table III. The consequent probabilistic schedule of
DR and ILC for different scenarios are given in Table IV.
As explained in Section III, in order to select the best solution
among the obtained Pareto optimal set, fuzzy satisfying method
is utilized here. It is evident from the last column of Table
II that the best solution is Solution#29, with the maximum
weakest membership function of 0.7890. The corresponding
CVC cost and LM are equal to $175161.218 and 0.2789,
respectively. Figure 5 gives the redispatch of fast-response
generating units for this case. Besides, the voltage magnitudes
of generator buses for both Solution#1 and Solution#29 are
given in Fig. 6. The active and reactive power output of wind
farms in this case, are given in Table V. Also, the resulting
DR and ILC schedules for this case are given in Table VI.
Active Power Schedule (MW)
in [27]. Also, in order to choose the best solution among the
obtained Pareto optimal set, fuzzy satisfying approach [31] is
adopted in this paper. This approach is described in [27].
0.25
0.2
0.15
0.1
0
Fig. 3.
1
2
3
4
5
Overal Cost ($)
6
7
8
9
5
x 10
The Pareto optimal front of the two objective functions
The expected cost for Solution#29 is $175161.218 , which
is much greater than the corresponding cost of $10431.511
for Solution#1. On the other hand, the LM in the former is
0.2789, which is greater than the 0.1000 in the latter. Also,
for Solution#29 the sum of expected DR and ILC schedules
are 6.29 MW and 188.29 MW, respectively. While these values
6
T HE PARETO OPTIMAL SOLUTIONS
Solution #(k)
f1 ($)
f2
f max −fk
1
f max −f min
1
1
f min −fk
2
f min −f max
2
2
Min
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
10431.511
10562.983
10563.115
10569.949
10580.599
10672.896
10748.059
11120.455
11801.023
12390.736
12406.505
12737.061
13112.559
18691.916
26448.570
30360.447
35470.611
36367.421
37324.050
37756.529
38426.818
58819.375
92731.830
121278.919
146924.208
154405.201
160232.543
174941.335
175161.218
192337.631
227002.052
286672.194
348531.071
346772.318
362514.944
383521.825
399504.032
443335.754
632271.786
841307.979
0.1000
0.1004
0.1118
0.1236
0.1354
0.1472
0.1589
0.1707
0.1800
0.1825
0.1837
0.1848
0.1856
0.1943
0.2061
0.2129
0.2179
0.2201
0.2210
0.2213
0.2217
0.2297
0.2415
0.2533
0.2651
0.2717
0.2741
0.2768
0.2789
0.2831
0.2886
0.3004
0.3122
0.3139
0.3158
0.3181
0.3192
0.3209
0.3240
0.3267
1.0000
0.9998
0.9998
0.9998
0.9998
0.9997
0.9996
0.9992
0.9984
0.9976
0.9976
0.9972
0.9968
0.9901
0.9807
0.9760
0.9699
0.9688
0.9676
0.9671
0.9663
0.9418
0.9009
0.8666
0.8357
0.8267
0.8197
0.8020
0.8017
0.7811
0.7393
0.6675
0.5931
0.5952
0.5763
0.5510
0.5317
0.4790
0.2516
0
0.0000
0.0017
0.0520
0.1040
0.1560
0.2080
0.2600
0.3120
0.3530
0.3640
0.3691
0.3739
0.3776
0.4160
0.4680
0.4980
0.5200
0.5296
0.5337
0.5352
0.5369
0.5720
0.6240
0.6760
0.7280
0.7575
0.7677
0.7800
0.7890
0.8076
0.8320
0.8840
0.9360
0.9437
0.9517
0.9618
0.9667
0.9743
0.9880
1
0.0000
0.0017
0.0520
0.1040
0.1560
0.2080
0.2600
0.3120
0.3530
0.3640
0.3691
0.3739
0.3776
0.4160
0.4680
0.4980
0.5200
0.5296
0.5337
0.5352
0.5369
0.5720
0.6240
0.6760
0.7280
0.7575
0.7677
0.7800
0.7890
0.7811
0.7393
0.6675
0.5931
0.5952
0.5763
0.5510
0.5317
0.4790
0.2516
0
Active Power Redispatch (MW)
TABLE II.
Active Power Redispatch (MW)
B46
B49
B54
B55
B56
B87
B89
B90
Generator Bus $umber
Fig. 4. Active power redispatch of fast-response generating units for Solution#1
are 0.79 MW and 6.18 MW, respectively, for Solution#1. The
considerable difference between the expected costs for these
two solutions, is mainly due to employing a large amount of
expensive ILC, to obtain the LM of 0.2789 in Solution#29.
TABLE III.
ACTIVE (M W ) AND REACTIVE (M V Ar) POWER
GENERATION OF WIND FARMS IN DIFFERENT SCENARIOS FOR S OLUTION #1
Qw
b,s
TABLE IV.
s8
s9
B47
1.7
1.7
B48
1
1
132.74
139.73
-28.23
37.81
150
s4
0
0
132.74
139.73
-28.23
37.81
150
s5
31.38
43.28
64.95
-48.72
94.35
31.69
131.19
s6
79.7
73.18
55.59
-47.43
96.15
35.22
121.96
s7
s8
s9
132.74
-50
101.9
37.81
150
3.01
30.06
27.15
24.21
44.33
-1.61
31.72
28.93
27.68
44.88
DR AND ILC SCHEDULES IN DIFFERENT SCENARIOS FOR
S OLUTION #1
DR
∆Pb,s
B51
0.85
0.85
(MW)
B57
0.6
0.6
B82
2.7
2.7
B88
2.4
2.4
B1
9.4
10.28
B2
19.15
20
B3
0
11.52
B7
0
11.66
ILC
(MW)
∆Pb,s
B11
B12
0
0
10.6
13.78
B13
15.03
28.27
B14
0
6.02
B41
0
3.19
B24
B25
B46
B49
B54
B55
B56
B87
B89
B90
Bus
B14
B51
B115
B14
B51
B57
B102
s1
140.12
0
0
-10.56
130.92
116.84
-45
s2
200
0
12.72
-15.74
131.1
118.47
-46.7
s3
200
0
102.77
-15.62
132.02
120.5
-42.97
s4
100
0
41.9
4.87
130.92
116.84
-43.79
s5
100
8.28
100
3.54
127.84
118.37
-46.24
s6
100
87.17
100
8.57
100.4
120.97
-43.7
s7
0
0
0
0.15
103.16
122
-50
s8
0
0
0
-5.05
109.06
122.3
-50
s9
0
0
0
-4.23
114.94
122.61
-49.36
literature to find them [35].
-250
s3
79.28
73.63
55.7
-47.43
96.15
38.34
121.85
0
Qw
b,s
-200
s2
31.38
43.28
64.95
-48.72
94.35
31.69
131.19
20
w
Pb,s
-150
s1
40
ACTIVE (M W ) AND REACTIVE (M V Ar) POWER
GENERATION OF WIND FARMS IN DIFFERENT SCENARIOS FOR
S OLUTION #29
-100
Bus
B14
B115
B14
B51
B57
B102
B115
60
TABLE V.
-50
w
Pb,s
80
Fig. 5. Active power redispatch of fast-response generating units for Solution#29 (MW)
0
B25
100
Generator Bus #umber
50
B24
120
B117
15.75
20
In this work, the locations of wind turbines in the grid are
priorly known. In case the optimal locations of wind turbines
are to be determined, there exist some efficient methods in the
A. Value of stochastic solution
It is obvious that using deterministic model results in simpler
formulation and lower problem size in comparison with the
stochastic models. In order to give an insight about the decisions
made by two methods, the following studies are carried out.
Expected Value (EV) solution: In this case all of the
random variables are replaced by the corresponding expected values (mean values of different scenarios) and the
resulting deterministic problem is solved. The obtained
objective function value is indicated as EV solution.
Stochastic solution (SS) or recourse problem (RP): the
stochastic problem is solved considering all of the scenarios. The obtained objective function is called RP.
Expected outcome of using the expected value (EEV): In
this case we have fixed the first stage variables with the
results obtained from deterministic case (i.e. the results
obtained from EV solution) and the stochastic program is
solved considering the scenarios. EEV represents the true
cost of the deterministic solution.
The value of stochastic solution (VSS) is calculated by subtracting the RP from the EEV as follows [36]:
V SS = EEV − RP
(40)
Table VII compares the obtained best compromise solution
using the three mentioned methods. The VSS for cost is
equal to $35,014.982 which indicates the extra cost of using
deterministic method instead of the stochastic model.
V. C ONCLUSION
In this paper, a probabilistic methodology is proposed for
corrective voltage control (CVC) of power systems. The proposed CVC aims to employ demand response (DR) along with
other resources as an effective tool to avoid voltage collapse
and provides a desired post-contingency loading margin (LM).
It considers the uncertainties associated with demand values and
wind power generation. The uncertainties are modeled using
scenario-based approach. The CVC problem is formulated as a
1.08
1.06
1.04
1.02
1
0.98
0.96
0.94
0.92
0.9
0.88
0.86
Solution#29
Solution#1
B1
B4
B6
B8
B12
B15
B18
B19
B24
B25
B26
B27
B31
B32
B34
B36
B40
B42
B46
B49
B54
B55
B56
B59
B61
B62
B65
B66
B69
B70
B72
B73
B74
B76
B77
B80
B85
B87
B89
B90
B91
B92
B99
B100
B103
B104
B105
B107
B110
B111
B112
B113
B116
Voltage Magnitude (pu)
7
Generator Bus Number
Fig. 6.
Voltage magnitude of generator buses (pu), for solutions #1 and #29
TABLE VI.
DR AND ILC SCHEDULES (MW) IN DIFFERENT SCENARIOS
FOR S OLUTION #29
Bus
B47
B48
B51
B57
B82
B88
B1
B2
B3
B6
B7
B11
B12
B13
B14
B16
B19
B20
B29
B33
B34
B36
B39
B41
B70
B74
B92
B93
B94
B95
B101
B102
B105
B110
B112
B117
B118
DR
∆Pb,s
ILC
∆Pb,s
TABLE VII.
s1
0
0
0
0
2.7
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
37.79
0
0
0
0
36.03
45.66
12
0
0
22
5
0
0
0
0
0
s2
1.7
1
0.85
0.6
2.7
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
32.87
0
0
0
0
37.58
44.42
12
0
11.6
22
5
0
0
0
0
0
s3
1.7
1
0.85
0.6
2.7
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
27.67
0
0
0
0
38.59
33.13
12
0
19.91
22
5
0
0
0
0
0
s4
0
0
0
0
2.7
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
38.64
0
0
0
0
35.83
42.8
12
0
0
22
5
0
0
0
0
0
s5
1.7
1
0.85
0.6
2.7
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
37.36
0
0
0
0
37.39
40.41
12
0
15.7
22
5
0
0
0
0
0
s6
1.7
1
0.85
0.6
2.7
2.4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
42.39
0
0
0
0
39.19
38.33
12
0
15.31
22
5
0
0
0
0
0
s7
1.7
1
0.85
0.6
2.7
2.4
0
14.01
9.47
0
0
6.82
10.75
24.43
0
0
0
0
0
0
45.05
0
0
0
5.29
33.23
22.93
12
7.36
42
22
5
0
0
0
20
12.78
s8
1.7
1
0.85
0.6
2.7
2.4
0
17.61
18.27
2.63
18.99
27.18
10.31
30.74
10.69
5.59
0
0
10.93
0
33.9
0
0
0
0
34.07
25.84
12
7.7
42
22
5
0
0
0.12
20
15.96
s9
1.7
1
0.85
0.6
2.7
2.4
0.39
20
22.8
0
18.92
39.13
17.55
34
14
9.74
4.26
1.75
15.05
8.49
25.64
0
6.65
8.12
0
34.91
25.98
12
9.48
42
22
5
1.06
0.25
13.18
20
19.3
T OTAL COST AND LOADING MARGINS USING DIFFERENT
UNCERTAINTY CONSIDERATION METHODS .
Method
Expected total cost
Expected loading margin
EV
1700191.9
0.2788
RP
175161.2
0.2789
EEV
210176.2
0.2889
multi-objective optimization problem. The objective functions
are satisfying a desired expected LM and minimization of its
corresponding expected CVC cost. This problem is solved using
ǫ-constraint technique, to achieve the corresponding Pareto
optimal set. Then, by using the Fuzzy satisfying method, the
best solution is selected among the optimal set. The proposed
approach is implemented on IEEE 118-bus test system, by
simulation of a double contingency case. Numerical results
show that in order to attain higher values of LM, more CVC
cost is imposed. Hence, the system operator should make fair
compromise between the desired LM and its corresponding
CVC cost. The presented results show the effectiveness of the
proposed probabilistic approach, to deal with the corrective
voltage control of power systems.
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Abbas Rabiee received the B.Sc. degree in electrical engineering from Iran
University of Science and Technology (IUST), Tehran, in 2006, and the M.Sc.
and Ph.D. degrees in electrical power engineering from Sharif University of
Technology (SUT), Tehran, Iran, in 2008 and 2013, respectively. Currently,
he is with the Department of Electrical Engineering, Faculty of Engineering,
University of Zanjan, Zanjan. Iran. His research interests include power system
operation and security, and the application of optimization techniques in power
system operation.
Alireza Soroudi (M14) Received the B.Sc. and M.Sc. degrees from Sharif
University of Technology, Tehran, Iran, in 2002 and 2004, respectively, both
in electrical engineering. and the joint Ph.D. degree from Sharif University
of Technology and the Grenoble Institute of Technology (Grenoble-INP),
Grenoble, France, in 2011. He is the winner of the ENRE Young Researcher
Prize at the INFORMS 2013.
Behnam Mohammadi-Ivatloo (S09, M12) received the B.Sc. degree in
electrical engineering from University of Tabriz, Tabriz, Iran, in 2006, the M.Sc.
and PhD degree from the Sharif University of Technology, Tehran, Iran, in 2008,
all with honors. He currently is an assistant professor with Faculty of Electrical
and Computer Engineering, University of Tabriz, Tabriz, Iran. His main area
of research is economics, operation and planning of intelligent energy systems
in a competitive market environment.
Mostafa Parniani (Senior Member, IEEE) is an associate professor at Sharif
University of Technology (SUT), Tehran, Iran. He received his B.Sc. degree
from Amirkabir University of Technology, Iran, in 1987, and the M.Sc. degree
from SUT in 1989, both in Electrical Power Engineering. He worked for
Ghods-Niroo Consulting Engineers Co. and for Electric Power Research Center
(EPRC) in Tehran during 1988-90. He obtained the Ph.D. degree in Electrical
Engineering from the University of Toronto, Canada, in 1995. He was a
visiting scholar at Rensselaer Polytechnic Institute, USA, during 2005-2006.
His research interests include power system dynamics and control, applications
of power electronics in power systems and renewable energies.